The sum of two numbers is $131$, and their difference is $47$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 131}$ ${x-y = 47}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 178 $ $ x = \dfrac{178}{2} $ ${x = 89}$ Now that you know ${x = 89}$ , plug it back into $ {x+y = 131}$ to find $y$ ${(89)}{ + y = 131}$ ${y = 42}$ You can also plug ${x = 89}$ into $ {x-y = 47}$ and get the same answer for $y$ ${(89)}{ - y = 47}$ ${y = 42}$ Therefore, the larger number is $89$, and the smaller number is $42$.